What is the best way to transfer the contents of my old 2010 iMac running High Sierra to my new Mac Mini running Mojave?

No, you are not correct.

The best way to transfer your content from an old to a new Mac would be to use Migration Assistant and a Time Machine backup of the data on the old Mac. This will NOT result in the installation of the old Mac OS onto the new computer. Only the user accounts and user data would be transferred.

Please see: Move content to a new Mac - Apple Support

Also see: Back up your Mac with Time Machine - Apple Support

One important thing to note since you are using Mojave, any apps that you are currently using could be 32 bit and will not run on the new MacMini. In addition, some apps may simply not work with the latest macOS Ventura.

You will need to do a bit of homework to determine which apps will work or require upgrades or even replacement with a different but similar app.

Yes, Mojave still runs 32-bit software. It is the last version of macOS with that capability.

Catalina cannot. Nor can Big Sur, Monterey or Ventura.

To learn which of your apps are 64-bit and compatible with the newer OSs you can use the Go64 utility app from St Claire Software. https://www.stclairsoft.com/Go64/

I thpught that Mojave still had the ability to run 32 bit apps. Is this true?

How new is your new Mac mini if it’s running Mojave?

Use Migration Assistant and an ethernet cable connected between the two machines.